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The formula for 'success'
« on: 2004-08-02 21:10:40 » |
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The formula for 'success'
John Allen Paulos
The Guardian, Thursday July 22, 2004
http://www.guardian.co.uk/life/lastword/story/0,,1265949,00.html
Pardon my exponents, writes John Allen Paulos, but mathematical laws that describe web links can also work for other complex systems
Might a discovery about the connectivity of the internet have relevance to power and wealth disparities in the world? A couple of years ago, Albert-Laszló Barabasi, a physicist at Notre Dame University, and two associates published a paper maintaining that websites link together in a way that accounts for the existence of relatively few very popular mega-sites and progressively more sites with fewer links. The "monkey-see-monkey-do" effect of many sites pointing to popular addresses leading proportionally more sites to do the same thing results in a so-called mathematical power law.
Barabasi showed, if you'll pardon my exponents, that the likelihood of a site having links to k other sites is roughly proportional to 1/k^3 - or inversely proportional to the third power of k. This means there are approximately one-eighth as many sites with 10 links as there are sites with five links since 1/10^3 is one-eighth of 1/5^3. In general, the number of sites with k links declines quickly as k increases. (The number is not described by the normal bell-shaped distribution which would result in even fewer sites with many links.)
So what, you say. Well, the power laws that characterise the internet seem to characterise many other complex systems. A number of scientists, including physicist Per Bak, who made an extensive study of them in his book How Nature Works, claim that such 1/k^m laws (for various exponents) are typical of many biological, geological, musical and economic processes. They tend to arise in a variety of situations in which "self-organisation" plays a role. Traffic jams, to cite an unrelated dynamic, often seem to obey a power law, with jams involving k cars occurring with a probability roughly proportional to 1/k^m for an appropriate ^m.
There is even a power law in linguistics. In English, for example, the word "the" appears most frequently and is said to have rank order 1; the words of rank 2, 3, and 4 are "of," "and," and "to," respectively. Zipf's Law relates the frequency of a word to its rank order k and states that a word's frequency in a written text is proportional to 1/k^1; that is, inversely proportional to the first power of k. (Thus "of " occurs half as frequently as "the", "and" a third as frequently as "the" - and "synecdoche" hardly at all.)
Sizes of cities, avalanches and earthquakes also follow power laws (with different exponents), and there is speculation that the number of books sold online at Amazon does, too. Mathematician Benoit Mandelbrot has shown that stock price movements follow a 1/k^m law, indicating that the trading network of investors, funds and brokerages is sometimes more herdlike than standard pictures acknowledge.
The flocking tendency of internet surfers, English speakers and market investors suggests a generalised "big-gets-bigger" phenomenon, which seems to be the source of some power laws. Much work remains to be done to understand why power laws are so pervasive. (The ubiquity of the bell distribution is well understood.)
Whatever their source, I think that more than a mathematical pun may connect power laws to economic, media and political power. Along various social dimensions, the dynamics underlying power laws may lead naturally to the development of a few large, powerful economic, media, and political elites and the disparities that result. Such stark inequalities seem to reign throughout the world.
The UN issued a report a few years ago saying the net worth of the world's three richest families - the Gateses, the Sultan of Brunei and the Waltons, of Wal-Mart - exceeded the GDP of the 43 poorest nations. The pattern holds within countries too. The ratio of remuneration of a US firm's chief to that of the average employee is at an all-time high of about 500.
Not only is there a strong tendency for the rich to get richer, but also for the healthy to get healthier. Contrast the money, energy and advertising that go into treatments for wrinkles, impotence, baldness and obesity with that for malaria, diarrhoea or tuberculosis. This disproportion is reflected in media, politics and the military. Disparities are no doubt necessary for complex societies to function, but they needn't be as extreme as they often are. A worrisome speculation is that market volatility, the shape of the internet and increasingly unipolar geopolitical power are indicators of even greater social disparities to come.
- John Allen Paulos is a professor of mathematics at Temple University, Philadelphia, and bestselling author of Innumeracy and A Mathematician Plays the Stock Market.
http://www.math.temple.edu/paulos
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